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arXiv · 2610.29620

Linear Stability of Double-Double Orbits in the Equal-Mass Four-Body Problem via Index Theory

Zijian Meng
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Abstract

In this paper, using the Maslov-type index theory in symplectic geometry, we study the linear stability of the double-double orbits in the planar equal-mass four body problem. By analyzing its symmetry properties and variational characterization, we clarify the relationship between linear stability and its variational nature on various loop spaces and show that for any rotation angle $θ=\fracπ{n}$, the retrograde double-double orbits are linearly stable with integer $n\in[3,30]$ and the prograde double-double orbits are linearly stable with integer $n\in[4,30]$. We also give the distributions of the eigenvalues on the unit circle of the monodromy matrices associated with the orbits. The basic idea is as follows: the variational characterization of the orbits provides information about its Morse index; based on its relation to the Maslov-type index, our stability criteria come into play.

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Zijian Meng. 2026-10-21. Linear Stability of Double-Double Orbits in the Equal-Mass Four-Body Problem via Index Theory. https://arxiv.org/abs/2610.29620

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